SOLVING EXPONENTIAL EQUATIONS WITH UNLIKE BASES

Solve the equation. Check your solution.

Problem 1 :

(1/5)^x = 125

Solution :

(1/5)^x = 125

Writing 125 in exponential form, we get

(1/5)^x = 5³

5^-x = 5³

Bases are equal. So, we can equate the powers.

-x = 3

x = -3

Check the solution :

(1/5)^-3 = 125

(1/5)^-3 = 5³

5³ = 5³

Problem 2 :

(1/4)^x = 256

Solution :

(1/4)^x = 256

Writing 256 as a multiple of 4.

(1/4)^x = 4⁴

4^-x = 4⁴

-x = 4

x = -4

Problem 3 :

1/128 = 2^{5x + 3}

Solution :

1/128 = 2^{5x + 3}

(128)^-1 = 2^{5x+ 3}

Writing 128 as multiple of 2.

(2⁷)^-1 = 2^{5x + 3}

Since we have power raised another power, we have to multiply the powers.

2^-7 = 2^{5x +3}

-7 = 5x + 3

-7 - 3 = 5x

5x = -10

Dividing 5 on each sides.

5/5x = -10/5

x = -2

Problem 4 :

3^{4x – 9} = 1/243

Solution :

3^{4x – 9} = 1/243

3^{4x – 9} = (243)^-1

Since we have power raised another power, we have to multiply the powers.

3^{4x – 9} = (3⁵)^-1

3^{4x – 9} = 3^-5

4x – 9 = -5

4x = -5 + 9

4x = 4

Dividing 4 on each sides.

x = 1

Problem 5 :

36^{-3x + 3} = (1/216)^{x + 1}

Solution :

36^{-3x + 3} = (1/216)^{x + 1}

36^{-3x + 3} = (216^-1)^{x + 1}

36^{-3x + 3} = 216^{-x – 1}

(6²)^{-3x + 3} = (6³)^{-x - 1}

6^{-6x + 6} = 6^{-3x – 3}

-6x + 6 = -3x – 3

-6x + 3x = -3 - 6

-3x = -9

Dividing -3 on each sides.

-3x/-3 = -9/-3

x = 3

Problem 6 :

(1/27)^{4 - x} = 9^{2x - 1}

Solution :

(1/27)^{4 - x} = 9^{2x - 1}

(27^-1)^{4 - x} = 9^{2x - 1}

(27)^{-4 + x} = 9^{2x - 1}

(3³)^{-4 + x} = (3²)^{2x - 1}

3^{-12 + 3x} = 3^{4x – 2}

-12 + 3x = 4x – 2

-12 + 2 = 4x – 3x

-10 = x

Problem 7 :

7^{x - 5} = 49^x

Solution :

7^{x - 5} = 49^x

7^{x - 5} = (7²)^x

7^{x - 5} = (7^2x)

x - 5 = 2x

-5 = 2x – x

-5 = x

x = -5

Problem 8 :

216^x = 6^{x + 10}

Solution :

216^x = 6^{x + 10}

(6³)^x = 6^{x + 10}

6^3x = 6^{x+ 10}

3x = x + 10

3x – x = 10

2x = 10

Dividing 2 on each sides.

2x/2 = 10/2

x = 5

Problem 9 :

64^{2x + 4} = 16^5x

Solution :

64^{2x + 4} = 16^5x

(4³)^{2x + 4} = (4²)^5x

4^{6x + 12} = 4^10x

6x + 12 = 10x

6x – 10x = -12

-4x = -12

Dividing -4 on each sides.

-4x/-4 = -12/-4

x = 3

Problem 10 :

27^x = 9^{x - 2}

Solution :

27^x = 9^{x - 2}

(3³)^x = (3²)^{x – 2}

3^3x = 3^{2x – 4}

3x = 2x – 4

3x - 2x = - 4

x = -4